確率統計 特論 (Probability and...
Transcript of 確率統計 特論 (Probability and...
確率統計 ⋅ 特論 (Probability and Statistics)
April 17, 2019
来嶋 秀治 (Shuji Kijima)
システム情報科学研究院 情報学部門
Dept. Informatics, ISEE
Todays topics
• what is probability?
• probability space
確率統計 ⋅ 特論(Probability & Statistics)
共通基礎科目
Preview
About this class
3Evaluation
• exercises (レポート課題) =: z (/10)
Almost every class (10回程度)
Submit hard copy at the end of class, or
email to [email protected] before then.
• midterm (中間試験): (early) Nov., =:x (/50)
• final (期末試験): late Jan or early Feb, =:y (/50)
• score = x+y (+z), something like that…
• passing mark = 60. No makeup exam (追試は無い)
supplemental option for creditnot mandatory, but recommend.
4Lecture
• Use slides and/or board.
• Language: speak in Japanese, write in English.
• mathematical backgrounds (undergraduate level)
e.g., calculus (微積分), linear algebra (線形代数), etc…
• Questions are welcome in English or Japanese. Do not hesitate!
• Check the web page!
http://tcs.inf.kyushu-u.ac.jp/~kijima/
you can find exercise, keywords, and schedule.
• Office hour: Wed(水曜日) 15:00—18:00 (W2, 7F, 741)
• Topics ⇒ see http://tcs.inf.kyushu-u.ac.jp/~kijima/
as much as possible… sorry.
5References
• 藤澤洋徳, 確率と統計, 現代基礎数学 13, 朝倉書店, 2006, 4000円.• 中田寿夫,内藤貫太,確率・統計,学術図書出版社,2017. • M. Mitzenmacher, E. Upfel, Probability and Computing: Randomized Algorithms and Probabilistic Analysis, Cambridge University Press, 2005. (邦訳) 小柴健史, 河内亮周, 確率と計算 --乱択アルゴリズムと確率的解析--, 共立出版, 2009.
[確率論]• 伏見正則, 確率と確率過程, 朝倉書店, 2002, 2800円[統計学]• 日本統計学会, 統計学 (統計検定1級対応), 東京図書, 2013. • 東大教養学部統計学教室編, 統計学入門, 東京大学出版会, 1991, 2940円.
Check the web page! http://www.tcs.inf.kyushu-u.ac.jp/~kijima/
Requirement (前提知識)6
Analysis (解析学)
• Differentiation (微分)
dd𝑥𝑥
log 𝑥𝑥 ?
Integration by parts (部分積分)
• ∫𝑎𝑎𝑏𝑏 𝑥𝑥 exp(−𝑥𝑥) d𝑥𝑥
Linear algebra (線形代数学)
• Bases (基底)
• Eigenvalues/diagonalization ( 固有空間/対角化)
What is Probability?
Examples
Ex 1. Monty Hall problem --- ask Marilyn8
You are given the choice of three doors: Behind on door is a car; behind the others goats.
You pick a door, say “A”. The host (Monty), who knows what's behind the doors,
opens another door, say “C”, which he knows has a goat. He then says to you, “Do you want to pick door “B”?”
QuestionIs it to your advantage to switch your choice?
図: wikipedia”モンティーホール問題”より
Ex 1. A modified Monty Hall problem --- ask Marilyn9
You are given the choice of 26 doors: Behind on door is a car; behind the others goats.
You pick a door, say “A”. The host (Monty), who knows what's behind the doors,
opens all other doors, say “C” to “Z”, except for “B”, each of which he knows has a goat.
He then says to you, "Do you want to pick door “B”?”
QuestionIs it to your advantage to switch your choice?
図: wikipedia”モンティーホール問題”より
Ex 2. Bertrand paradox10
Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random.
QuestionWhat is the probability that the chord is longer than a side of the triangle?
Ex 2. Bertrand paradox11
Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random.
QuestionWhat is the probability that the chord is longer than a side of the triangle (=:x)?
Answer 1: The "random midpoint" method.
Choose a point anywhere within the circle, and
construct a chord with the chosen point as its midpoint.
The chord is longer than x
iff the chosen point within a small circle.
⇒ the probability is 1/4
Ex 2. Bertrand paradox12
Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random.
QuestionWhat is the probability that the chord is longer than a side of the triangle (=:x)?
Answer 2: The "random radius" method:
Choose a radius of the circle and a point on the radius.
the chord through this point and perpendicular to the radius.
The chord is longer than x
iff the chosen point on a blue line.
⇒ the probability is 1/2
Answer 3: The "random endpoints" method::
Choose two random points on the circumference of the circle and
draw the chord joining them.
The chord is longer than x
iff π/3 ≤ θ ≤ 2π/3
⇒ the probability is 1/3
Ex 2. Bertrand paradox13
Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random.
QuestionWhat is the probability that the chord is longer than a side of the triangle (=:x)?
θ
Ex 2. Bertrand paradox14
Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random.
QuestionWhat is the probability that the chord is longer than a side of the triangle (=:x)?
θ
What does“a chord of the circle is chosen at random” mean?
Probability Space
Definitions
Axiom
Terminology
16
A probability space is defined by (Ω,ℱ,𝑃𝑃)Ω: sample space(標本空間); a set of elementally events(標本点), an event(事象) is a subset of Ω.
ℱ: σ-algebra (⊆ 2Ω); a set of events. 𝑃𝑃: probability measure(確率測度); a function ℱ → ℝ, probability of an event.
Definition: Probability Space
ℱ is a σ-algebra
1. ℱ contains the empty set: ∅ ∈ ℱ
2. ℱ is closed under complements: 𝐴𝐴 ∈ ℱ ⇒ �𝐴𝐴 ∈ ℱ
3. ℱ is closed under countable unions: 𝐴𝐴𝑖𝑖 ∈ ℱ⇒ ⋃𝑖𝑖 𝐴𝐴𝑖𝑖 ∈ ℱ
17Definition: Probability Space
𝑃𝑃 is a probability measure (Kolmogorov axioms)
1. 𝑃𝑃 𝐴𝐴 ≥ 0 for any 𝐴𝐴 ∈ ℱ.
2. 𝑃𝑃 Ω = 1.3. Any countable sequence of mutual exclusive events 𝐴𝐴1,𝐴𝐴2, …
satisfies that 𝑃𝑃 𝐴𝐴1 ∪ 𝐴𝐴2 ∪ ⋯ = 𝑃𝑃 𝐴𝐴1 + 𝑃𝑃 𝐴𝐴2 + ⋯.
A probability space is defined by (Ω,ℱ,𝑃𝑃)Ω: sample space(標本空間); a set of elementally events(標本点), an event(事象) is a subset of Ω.
ℱ: σ-algebra (⊆ 2Ω); a set of events. 𝑃𝑃: probability measure(確率測度); a function ℱ → ℝ, probability of an event.
18Example: Probability Space
Ex. 1. Die
Ω = {1,2,3,4,5,6},
ℱ = 2Ω
𝑃𝑃 𝐴𝐴 = 𝐴𝐴6
for any 𝐴𝐴 ⊆ Ω.
A probability space is defined by (Ω,ℱ,𝑃𝑃)Ω: sample space(標本空間); a set of elementally events(標本点), an event(事象) is a subset of Ω.
ℱ: σ-algebra (⊆ 2Ω); a set of events. 𝑃𝑃: probability measure(確率測度); a function ℱ → ℝ, probability of an event.
19Example: Probability Space
Ex. 2. tossing a fair coinTossing a coin. Ω = ℎ, 𝑡𝑡 𝐹𝐹 = 2 ℎ,𝑡𝑡
𝑃𝑃 ∅ = 0, 𝑃𝑃 ℎ = 1/2 , 𝑃𝑃 𝑡𝑡 = 1/2, 𝑃𝑃 ℎ, 𝑡𝑡 = 1
A probability space is defined by (Ω,ℱ,𝑃𝑃)Ω: sample space(標本空間); a set of elementally events(標本点), an event(事象) is a subset of Ω.
ℱ: σ-algebra (⊆ 2Ω); a set of events. 𝑃𝑃: probability measure(確率測度); a function ℱ → ℝ, probability of an event.
20Example: Probability Space
Ex. 2. tossing 2 fair coinsTossing 2 coins, which look alike. We can observe only how many heads or tails are. Ω = ℎ,ℎ , ℎ, 𝑡𝑡 , 𝑡𝑡,ℎ , 𝑡𝑡, 𝑡𝑡 (why?) 𝐹𝐹 = 2 0,1,2 (why?) 𝑃𝑃 ∅ = 0, 𝑃𝑃 0 = 1/4 , 𝑃𝑃 1 = 2/4, 𝑃𝑃 {0,1} = 3/4, …
A probability space is defined by (Ω,ℱ,𝑃𝑃)Ω: sample space(標本空間); a set of elementally events(標本点), an event(事象) is a subset of Ω.
ℱ: σ-algebra (⊆ 2Ω); a set of events. 𝑃𝑃: probability measure(確率測度); a function ℱ → ℝ, probability of an event.
21Example: Probability Space
Ex. 2. tossing 3 fair coinsTossing 3 coins, which look alike. We can observe only how many heads or tails are.
Ω = ℎ,ℎ,ℎ , ℎ,ℎ, 𝑡𝑡 , ℎ, 𝑡𝑡,ℎ , ℎ, 𝑡𝑡, 𝑡𝑡 ,𝑡𝑡,ℎ,ℎ , 𝑡𝑡,ℎ, 𝑡𝑡 , 𝑡𝑡, 𝑡𝑡,ℎ , (𝑡𝑡, 𝑡𝑡, 𝑡𝑡) (why?)
𝐹𝐹 = 2 0,1,2,3 (why?) 𝑃𝑃 ∅ = 0, 𝑃𝑃 0 = 1/8 , 𝑃𝑃 1 = 3/8, 𝑃𝑃 #ℎ is odd = 1/2, …
A probability space is defined by (Ω,ℱ,𝑃𝑃)Ω: sample space(標本空間); a set of elementally events(標本点), an event(事象) is a subset of Ω.
ℱ: σ-algebra (⊆ 2Ω); a set of events. 𝑃𝑃: probability measure(確率測度); a function ℱ → ℝ, probability of an event.
22Property of Probability (1/2)Kolmogolov’s axiom (P1), (P2), (P3) implies the following.
Thm. 1
Let Ω,ℱ,𝑃𝑃 be an arbitrary probability space.
(1) 𝑃𝑃 ∅ = 0.
(2) If 𝐴𝐴 ⊆ 𝐵𝐵 (𝐴𝐴,𝐵𝐵 ∈ ℱ), then 𝑃𝑃 𝐵𝐵 ∖ 𝐴𝐴 = 𝑃𝑃 𝐵𝐵 − 𝑃𝑃 𝐴𝐴 .(rem. 𝐵𝐵 ∖ 𝐴𝐴 = 𝐵𝐵 ∩ �𝐴𝐴 )
(3) If 𝐴𝐴 ⊆ 𝐵𝐵 (𝐴𝐴,𝐵𝐵 ∈ ℱ), then 𝑃𝑃 𝐴𝐴 ≤ 𝑃𝑃 𝐵𝐵 .
(4) 𝑃𝑃 𝐴𝐴 ≤ 1 for any 𝐴𝐴 ∈ ℱ.
(5) 𝑃𝑃 �𝐴𝐴 = 1 − 𝑃𝑃(𝐴𝐴) for any 𝐴𝐴 ∈ ℱ.
23Property of Probability (2/2)
Thm. 1. (cont.)
(6) P 𝑃𝑃 𝐴𝐴 ∪ B = 𝑃𝑃 𝐴𝐴 + 𝑃𝑃 𝐵𝐵 − 𝑃𝑃 𝐴𝐴 ∩ 𝐵𝐵 for any 𝐴𝐴 ∈ ℱ.
(7) If 𝐴𝐴1,𝐴𝐴2, … ∈ ℱ satisfy 𝐴𝐴1 ⊆ 𝐴𝐴2 ⊆ ⋯,
then 𝑃𝑃 ⋃𝑖𝑖 𝐴𝐴𝑖𝑖 = lim𝑛𝑛→∞
𝑃𝑃 𝐴𝐴𝑛𝑛
(8) If 𝐴𝐴1,𝐴𝐴2, … ∈ ℱ satisfy 𝐴𝐴1 ⊇ 𝐴𝐴2 ⊇ ⋯,
then 𝑃𝑃 ⋂𝑖𝑖 𝐴𝐴𝑖𝑖 = lim𝑛𝑛→∞
𝑃𝑃 𝐴𝐴𝑛𝑛
(9) For any 𝐴𝐴1,𝐴𝐴2, … ∈ ℱ,
𝑃𝑃 ⋃𝑖𝑖 𝐴𝐴𝑖𝑖 = 0 ⇔ ∀𝑖𝑖 ≥ 0,𝑃𝑃 𝐴𝐴𝑖𝑖 = 0
Kolmogolov’s axiom (P1), (P2), (P3) implies the following.